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Chapter 1: Problem 19

Sketch the region given by the set. $$\\{(x, y) | y=2\\}$$

Short Answer

Expert verified
The region is a horizontal line at \( y = 2 \).

Step by step solution

01

Understand the Given Condition

The condition \( y = 2 \) specifies that for all points \((x, y)\) in the plane, the \(y\)-coordinate must be exactly \(2\). This means any \(x\)-coordinate is allowed as long as the \(y\) remains \(2\).
02

Interpret the Geometric Representation

The equation \( y = 2 \) represents a horizontal line in the coordinate plane. It is parallel to the x-axis and passes through the point \((0, 2)\).
03

Sketch the Line

On a graph, draw a straight line horizontally across the plane. Ensure this line passes through the point \((0, 2)\) on the y-axis. Extend the line in both directions along the x-axis to cover all possible \(x\)-values.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Horizontal Line
A horizontal line is a fundamental concept in coordinate geometry, characterized by having a constant y-coordinate across all its points. In our example with the equation \( y = 2 \), every point on this line shares the same y-value of 2, irrespective of the x-coordinate.
  • A horizontal line runs left to right, parallel to the x-axis in a coordinate plane.
  • It never slants upwards or downwards because there is no change in the y-value.
  • The slope of a horizontal line is always zero since there is no vertical change as you move along the line.
Understanding horizontal lines helps in recognizing patterns and solving geometric problems easily. If you remember that the y-value stays constant, you can quickly identify or sketch horizontal lines on a graph.
Graphing
Graphing is the process of plotting points, lines, and curves on a coordinate plane to visually represent mathematical concepts and relationships. It involves a systematic method of drawing elements based on their equations.
To graph a horizontal line like \( y = 2 \):
  • Identify the y-coordinate specified by the equation, which here is 2.
  • Mark a point on the y-axis at \((0,2)\).
  • Using this point as a reference, draw a line extending horizontally across the plane.
  • Ensure the line is straight and parallel to the x-axis, covering all potential x-values from negative to positive infinity.
Graphing serves as a valuable tool in understanding and analyzing mathematical representations visually, making abstract concepts more tangible.
Coordinate Plane
The coordinate plane is a two-dimensional surface where graphs and geometric shapes are plotted. It is defined by two perpendicular lines or axes: the x-axis (horizontal) and the y-axis (vertical). The plane is divided into four quadrants, each identified based on the sign of the x and y coordinates.
  • The point where the axes intersect is known as the origin, with coordinates \((0,0)\).
  • In our case, any point of the form \((x, 2)\) will lie on the horizontal line \( y = 2 \).
  • Understanding the layout of the coordinate plane is crucial for interpreting and sketching graphs accurately.
  • Each point on the plane is defined by an ordered pair \((x, y)\).
A solid grasp of the coordinate plane's structure allows you to effectively plot points and analyze relationships between different lines and curves.

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Most popular questions from this chapter

Factor the expression completely. $$3 x^{3}+5 x^{2}-6 x-10$$

Factor the expression by grouping terms. $$x^{3}+x^{2}+x+1$$

A trinomial of the form \(x^{4}+a x^{2}+b\) can sometimes be factored easily. For example, \(x^{4}+3 x^{2}-4=\left(x^{2}+4\right)\left(x^{2}-1\right) .\) But \(x^{4}+3 x^{2}+4\) cannot be factored in this way. Instead, we can use the following method. $$\begin{aligned}x^{4}+3 x^{2}+4 &=\left(x^{4}+4 x^{2}+4\right)-x^{2} \\\&=\left(x^{2}+2\right)^{2}-x^{2} \\\&=\left[\left(x^{2}+2\right)-x\right]\left[\left(x^{2}+2\right)+x\right] \\\&=\left(x^{2}-x+2\right)\left(x^{2}+x+2\right)\end{aligned}$$

Show that the equation represents a circle, and find the center and radius of the circle. $$x^{2}+y^{2}-\frac{1}{2} x+\frac{1}{2} y=\frac{1}{8}$$

Factor the expression completely. $$\left(1+\frac{1}{x}\right)^{2}-\left(1-\frac{1}{x}\right)^{2}$$
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